\f\f\f\f\f\f\f\f\f\f\f\f\f\f\f\f\f\f1) Option C is correct. They don't intersect as they don't have solution ,Hence parallel lines 2) 2 solutions are possible here as 2 unknowns and 2 equation are there 3) Put (7/5 , 3 ) in both equations to check if this is solution a) LHS= 15(7/5)+20= 41 AND RHS= 44-3=41 So this satisifies b) LHS=5(7/5)-7(3)= -14 and LHS =4 not equal to LHS ,hence not a solution 4) Ans : A) The solution is (0,-4) it's intersecting point Graph method 10 8 6 4 Y Axis 2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 -2 -4 X Axis 4) 5) ans Option B graph is correct. Parralel lines as slope m1 and m2 are same m1=m2= -1 There is no solution 6)Ans Option B: there are infintely many solutions as equation 2 is same as 1, divide equation (2) by 3 , you get equation 1 . which are same Graph method 10 8 6 4 Y Axis 2 0 0 -2 -4 0.5 1 1.5 2 2.5 X Axis 3 3.5 4 4.5 7)Ans: substitute y=3x in equation (1) , we have 2x+3x=5 , 5x=5 thus x=1 , put this in equation 2 We get y=3 Thus (1, 3) is solution 8) Ans : Put y=4x-3 in equation (1), we get -x - 4(4x-3)= -5 -17x= -17 thus x=1 , put this in equation 1 , we get Y=1, Thus (1,1) is solution 9) Ans: Put equation (2) in equation (1), we have 3y +10 =(3/2)y ; Thus y= -20/3 , put this in eq n (2) X=-10 Thus ( -10, -20/3) is solution 10) Ans\" Put equation (1) in equation (2), we have 6x-3(2x)=0 thus 0=0 Equation 1 and 2 are same , divide equation (2) by 2 , we get y=2x same to equation 1. So there are infinite solution to this 11)Ans: Substitute x=6y in equation 2 , We get 24y-24y=4 , i.e. 0 = 4 , impossible Hence no solution possible 12) Ans Mulitiply equation 2 by 3 We have 9x+12y=48 Subtract equation 1 from equation 2 , We get , 5y=20 , thus y=4 , Put this in equation 1, we get , 9x+28=28 , Thus x=0 (0,4) is the solution Put this (0,4 ) in both equation, both satisfies . Hence checked, this is the solution 13) Ans : Multiply equation 1 by 2 ,we get 4s+10t =16 Now Add equation 1 from equation 2 ,we get 5s=39 , thus s= 5/39 Put this in equation 1 , we get 10/39 +5t=8 Thus t= 1.5487 Thus ( 5/39 , 1.5487) is solution Put this ( 5/39 , 1.5487) in both equation, both satisfies . Hence checked, this is the solution 14) Ans : Multiply equation 1 by 2/5 and equation 2 by 1/3 , we get 2/15 (x) - 1/30 (y) =2 .....(3) 2/15(x) +2/9 (Y) =2 .....(4) Subtract equation 2 from equation 1, We get y=0 Put this y=0 in equation 3 to get x , X=15 , So (15, 0 ) is the solution Put this ( 15,0) in both equation, both satisfies . Hence checked, this is the solution 15) Ans: (2,3) is intersection point , hence this is solution Graph method 10 8.67 8 6 4 Y Axis 2 0 0 -2 -2 -0.67 0.5 0 1 1.5 2 -4 X Axis 2.5 3 3.5 4 4.5 16)Ans they are parallel lines( option B) , hence inconsistent equation s, hence no solution (option C) as they don't intersect Graph method 10 8 6 4 Y Axis 2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 -2 -4 X Axis 17 ANs : Divide equation 2 by -9 we get 8x-3y=6 , which is equal to equation 1 , It is the same line (Option B) , hence no solution (option C /)possible with two unknowns and 1 equation 18) Ans let small no. be x and big number be y Now x+y =103 ..... equation 1 And y- 3x =7 Subtract equation 2 from 1 , we get 4x=96 , thus x= 24 , Thus y= 103-24 =89 (24, 89 ) is solution Small number is 24 and larger one is 89 19) Ans Let heavy equi[ment operators be x and general laborers be y , Thus x+y= 37 ......equation (1) And 120x+86y=4304 ......equation (2) Multiply equation 1 by120 , we get 120x+120y=4440 Subtract equation 2 from this , we get 34y=136 , thus y=4 Now , x= 37-4 = 33 , Thus heavy equi[ment operators are 33 and general laborers are 4 20) ANs : Let regular seat be x and sleeper seat be y in number Thus x+y =96 ......equation (1) And 118x+295y=21063 ......equation (2) Multiply equation 1 by 118 , we get 118x+ 118y = 11328 , Subtract this from equation 2 we get 177y= 9735 thus y= 55 X=96-55 = 41 regular seat are 41 and sleeper seat be are 55 in no. 21) ANs : Let x be adults and y be children 3x+4y =79 multiply this by 2 , we have 6x+8y=158 2x+3y=56 multiply this by 3 , we have 6x+9y=168 Subtract 1 from 2 , we have Y=10 , Thus put y=10 in equation 1 , we have 3x=79-40=39 Thus x= 13 We have 13$ for adults and 10 dollars for children 22) ANs: Let x be donuts and y be coffees 4x+5y=7.36 multiply by 6 , we have 24x+30y=44.16 6x+3y=6.9 multiply by 4 , we have 24x+12y=27.6 Subtract 2 from 1 , We have 18y= 16.56 Y=0.92 Now ,put in equation 1 , we have x=0.69 Cost of donut is 0.69$ and coffee is 0.92$ 23) Ans: let speed of plane be x and of wind be y The solutions are: x=3; y=1/2 Explanation STEP 1: Multiply each equation by a number that will create opposite coefficients for x. In this case we will multiply first equation by -5 and the second equation by 7. After multiplying we have the following system: 350x+350y=875; 350x+350y =1225 STEP 2: add the two equations together to eliminate x from the system. 700y=350 STEP 3: find y y=1/2 STEP 4: substitute the value for y into the original equation to solve for x. 70x70(12)=175 x=3 24) Ans : x+y =32 2x+y =51 The solutions are: x=19; y=13 Explanation STEP 1: Multiply first equation by -2. After multiplying we have the following system: 2x2y =64 2x+y =51 STEP 2: add the two equations together to eliminate x from the system. y=13 STEP 3: find y y=13 STEP 4: substitute the value for y into the original equation to solve for x. x+(13)=32 x=19 Thus 2 pointers are 19 and free throws are 13 25) Ans : Let calls on weekdays be x and on weekend be y x+y=593; 2/25x+9/100y =2575 The solutions are: X=197; y=396 Explanation STEP 1: Get rid of the fractions. by multiplying second equation by 100. After multiplying we have the following system: x+y=593; 8x+9y =5140 STEP 2: Multiply first equation by -8. After multiplying we have the following system: 8x8y=4744; 8x+9y =5140 STEP 3: add the two equations together to eliminate x from the system. y=396 STEP 4: find y y=396 STEP 5: substitute the value for y into the original equation to solve for x. x+(396)=593 x=197 Thus 197 calls on weekdays and 396 calls on weekends 26) Ans : Let games won be x and lost be y . x+y =70; xy=32 The solutions are: x=51; y=19 Explanation STEP 1: add the two equations together to eliminate y from the system. 2x=102 STEP 2: find x x=51 STEP 3: substitute the value for x into the original equation to solve for y. (51)+y=70 y=19 Games won are 51 and lost are 19 in total 27) Ans : Let x be length and y be breadth , 2x+2y =24; xy=2 The solutions are: x=7, y=5 Explanation STEP 1: Multiply second equation by -2. After multiplying we have the following system: 2x+2y=24; 2x+2y =4 STEP 2: add the two equations together to eliminate x from the system. 4y=20 STEP 3: find y y=5 STEP 4: substitute the value for y into the original equation to solve for x. 2x+2(5)=24 x=7 Thus length is 7 and breadth is 5 units 28) ANs : Let x be hot dog and y be hamburger , Now 7x+2y=232; 2x7y =714 The solutions are: x=11653; y=405212 Explanation STEP 1: Get rid of the fractions.by multiplying first equation by 2 and second equation by 4. After multiplying we have the following system: 14x+4y=23 8x28y =71 STEP 2: Multiply first equation by 7. After multiplying we have the following system: 98x+28y=161 8x28y =71 STEP 3: add the two equations together to eliminate y from the system. 106x=232 STEP 4: find x x=116/53=2.1887 STEP 5: substitute the value for x into the original equation to solve for y. 7(11653)+2y=232 y=405/212= 1.9104 thus hot dogs are 2 and hamburger are 2 29) ans : let x be alcohol and y be water in ml x+y=620; x=93/5 The solutions are: x=93/5 = 18.6 ; y=3007/5=601.4 Explanation STEP 1: Get rid of the fractions. by multiplying second equation by 5. After multiplying we have the following system: x+y=620; 5x =93 STEP 2: Use second equation to find x 5x=93x=93/5 STEP 3: Substitute x=93/5 into the first equation to solve for y. 93/5+y=620y=3007/5 Thus alcohol is 18.6 ml and water is 601.4 ml 30) Ans : Let x be quantity of one solution and y be of the second solution , now x+y=10; 3/20x+1/20y =11/10 The solutions are: x=6; y=4 Explanation STEP 1: Get rid of the fractions. by multiplying second equation by 20. After multiplying we have the following system: x+y =10; 3x+y =22 STEP 2: Multiply first equation by -3. After multiplying we have the following system: 3x3y =30; 3x+y =22 STEP 3: add the two equations together to eliminate x from the system. 2y=8 STEP 4: find y y=4 STEP 5: substitute the value for y into the original equation to solve for x. x+(4)=10 x=6 Thus 6 litre of 1st solution and 4 litre of wnd solution is needed 31) Ans: Let 1st investment is x and 2nd one is y Now ; x+y=5900; 11/100x+9/100y =593 The solutions are: x=3100; y=2800 Explanation STEP 1: Get rid of the fractions. by multiplying second equation by 100. After multiplying we have the following system: x+y=5900; 11x+9y =59300 STEP 2: Multiply first equation by -11. After multiplying we have the following system: 11x11y=64900; 11x+9y =59300 STEP 3: add the two equations together to eliminate x from the system. 2y=5600 STEP 4: find y y=2800 STEP 5: substitute the value for y into the original equation to solve for x. x+(2800)=5900 x=3100 Thus 1st investment is 3100$ and 2nd one is 2800 $ 32) Ans: OPTION B: Explaination : If the problem includes where it is equal, then you will have a solid boundary line. In other words, if you have < or > , you will have a solid line for your boundary line. Here it is < or > . Henxe solid line . 33) Ans: Yes Option A IS CORRECT . also one can check from given graph . The point (3,-4) lies in green area, so this is in the region 34) ANs : 2x-y<= 3 and x+y <=8 The colored region is solution 35) Ans : 1) Option C is correct. They don't intersect as they don't have solution ,Hence parallel lines 2) 2 solutions are possible here as 2 unknowns and 2 equation are there 3) Put (7/5 , 3 ) in both equations to check if this is solution a) LHS= 15(7/5)+20= 41 AND RHS= 44-3=41 So this satisifies b) LHS=5(7/5)-7(3)= -14 and LHS =4 not equal to LHS ,hence not a solution 4) Ans : A) The solution is (0,-4) it's intersecting point Graph method 10 8 6 4 Y Axis 2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 -2 -4 X Axis 4) 5) ans Option B graph is correct. Parralel lines as slope m1 and m2 are same m1=m2= -1 There is no solution 6)Ans Option B: there are infintely many solutions as equation 2 is same as 1, divide equation (2) by 3 , you get equation 1 . which are same Graph method 10 8 6 4 Y Axis 2 0 0 -2 -4 0.5 1 1.5 2 2.5 X Axis 3 3.5 4 4.5 7)Ans: substitute y=3x in equation (1) , we have 2x+3x=5 , 5x=5 thus x=1 , put this in equation 2 We get y=3 Thus (1, 3) is solution 8) Ans : Put y=4x-3 in equation (1), we get -x - 4(4x-3)= -5 -17x= -17 thus x=1 , put this in equation 1 , we get Y=1, Thus (1,1) is solution 9) Ans: Put equation (2) in equation (1), we have 3y +10 =(3/2)y ; Thus y= -20/3 , put this in eqn (2) X=-10 Thus ( -10, -20/3) is solution 10) Ans\" Put equation (1) in equation (2), we have 6x-3(2x)=0 thus 0=0 Equation 1 and 2 are same , divide equation (2) by 2 , we get y=2x same to equation 1. So there are infinite solution to this 11)Ans: Substitute x=6y in equation 2 , We get 24y-24y=4 , i.e. 0 = 4 , impossible Hence no solution possible 12) Ans Mulitiply equation 2 by 3 We have 9x+12y=48 Subtract equation 1 from equation 2 , We get , 5y=20 , thus y=4 , Put this in equation 1, we get , 9x+28=28 , Thus x=0 (0,4) is the solution Put this (0,4 ) in both equation, both satisfies . Hence checked, this is the solution 13) Ans : Multiply equation 1 by 2 ,we get 4s+10t =16 Now Add equation 1 from equation 2 ,we get 5s=39 , thus s= 5/39 Put this in equation 1 , we get 10/39 +5t=8 Thus t= 1.5487 Thus ( 5/39 , 1.5487) is solution Put this ( 5/39 , 1.5487) in both equation, both satisfies . Hence checked, this is the solution 14) Ans : Multiply equation 1 by 2/5 and equation 2 by 1/3 , we get 2/15 (x) - 1/30 (y) =2 2/15(x) +2/9 (Y) =2 .....(3) .....(4) Subtract equation 2 from equation 1, We get y=0 Put this y=0 in equation 3 to get x , X=15 , So (15, 0 ) is the solution Put this ( 15,0) in both equation, both satisfies . Hence checked, this is the solution 15) Ans: (2,3) is intersection point , hence this is solution Graph method 10 8.67 8 6 4 Y Axis 2 0 0 -2 -2 -0.67 0.5 0 1 1.5 2 2.5 3 3.5 4 4.5 -4 X Axis 16)Ans they are parallel lines( option B) , hence inconsistent equation s, hence no solution (option C) as they don't intersect Graph method 10 8 6 4 Y Axis 2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 -2 -4 X Axis 17 ANs : Divide equation 2 by -9 we get 8x-3y=6 , which is equal to equation 1 , It is the same line (Option B) , hence no solution (option C /)possible with two unknowns and 1 equation 18) Ans let small no. be x and big number be y Now x+y =103 ..... equation 1 And y- 3x =7 Subtract equation 2 from 1 , we get 4x=96 , thus x= 24 , Thus y= 103-24 =89 (24, 89 ) is solution Small number is 24 and larger one is 89 19) Ans Let heavy equi[ment operators be x and general laborers be y , Thus x+y= 37 ......equation (1) And 120x+86y=4304 ......equation (2) Multiply equation 1 by120 , we get 120x+120y=4440 Subtract equation 2 from this , we get 34y=136 , thus y=4 Now , x= 37-4 = 33 , Thus heavy equi[ment operators are 33 and general laborers are 4 20) ANs : Let regular seat be x and sleeper seat be y in number Thus x+y =96 ......equation (1) And 118x+295y=21063 ......equation (2) Multiply equation 1 by 118 , we get 118x+ 118y = 11328 , Subtract this from equation 2 we get 177y= 9735 thus y= 55 X=96-55 = 41 regular seat are 41 and sleeper seat be are 55 in no. 21) ANs : Let x be adults and y be children 3x+4y =79 multiply this by 2 , we have 6x+8y=158 2x+3y=56 6x+9y=168 multiply this by 3 , we have Subtract 1 from 2 , we have Y=10 , Thus put y=10 in equation 1 , we have 3x=79-40=39 Thus x= 13 We have 13$ for adults and 10 dollars for children 22) ANs: Let x be donuts and y be coffees 4x+5y=7.36 multiply by 6 , we have 24x+30y=44.16 6x+3y=6.9 multiply by 4 , we have 24x+12y=27.6 Subtract 2 from 1 , We have 18y= 16.56 Y=0.92 Now ,put in equation 1 , we have x=0.69 Cost of donut is 0.69$ and coffee is 0.92$ 23) Ans: let speed of plane be x and of wind be y The solutions are: x=3; y=1/2 Explanation STEP 1: Multiply each equation by a number that will create opposite coefficients for x. In this case we will multiply first equation by -5 and the second equation by 7. After multiplying we have the following system: 350x+350y=875; 350x+350y =1225 STEP 2: add the two equations together to eliminate x from the system. 700y=350 STEP 3: find y y=1/2 STEP 4: substitute the value for y into the original equation to solve for x. 70x70(12)=175 x=3 24) Ans : x+y =32 2x+y =51 The solutions are: x=19; y=13 Explanation STEP 1: Multiply first equation by -2. After multiplying we have the following system: 2x2y =64 2x+y =51 STEP 2: add the two equations together to eliminate x from the system. y=13 STEP 3: find y y=13 STEP 4: substitute the value for y into the original equation to solve for x. x+(13)=32 x=19 Thus 2 pointers are 19 and free throws are 13 25) Ans : Let calls on weekdays be x and on weekend be y x+y=593; 2/25x+9/100y =2575 The solutions are: X=197; y=396 Explanation STEP 1: Get rid of the fractions. by multiplying second equation by 100. After multiplying we have the following system: x+y=593; 8x+9y =5140 STEP 2: Multiply first equation by -8. After multiplying we have the following system: 8x8y=4744; 8x+9y =5140 STEP 3: add the two equations together to eliminate x from the system. y=396 STEP 4: find y y=396 STEP 5: substitute the value for y into the original equation to solve for x. x+(396)=593 x=197 Thus 197 calls on weekdays and 396 calls on weekends 26) Ans : Let games won be x and lost be y . x+y =70; xy=32 The solutions are: x=51; y=19 Explanation STEP 1: add the two equations together to eliminate y from the system. 2x=102 STEP 2: find x x=51 STEP 3: substitute the value for x into the original equation to solve for y. (51)+y=70 y=19 Games won are 51 and lost are 19 in total 27) Ans : Let x be length and y be breadth , 2x+2y =24; xy=2 The solutions are: x=7, y=5 Explanation STEP 1: Multiply second equation by -2. After multiplying we have the following system: 2x+2y=24; 2x+2y =4 STEP 2: add the two equations together to eliminate x from the system. 4y=20 STEP 3: find y y=5 STEP 4: substitute the value for y into the original equation to solve for x. 2x+2(5)=24 x=7 Thus length is 7 and breadth is 5 units 28) ANs : Let x be hot dog and y be hamburger , Now 7x+2y=232; 2x7y =714 The solutions are: x=11653; y=405212 Explanation STEP 1: Get rid of the fractions.by multiplying first equation by 2 and second equation by 4. After multiplying we have the following system: 14x+4y=23 8x28y =71 STEP 2: Multiply first equation by 7. After multiplying we have the following system: 98x+28y=161 8x28y =71 STEP 3: add the two equations together to eliminate y from the system. 106x=232 STEP 4: find x x=116/53=2.1887 STEP 5: substitute the value for x into the original equation to solve for y. 7(11653)+2y=232 y=405/212= 1.9104 thus hot dogs are 2 and hamburger are 2 29) ans : let x be alcohol and y be water in ml x+y=620; x=93/5 The solutions are: x=93/5 = 18.6 ; y=3007/5=601.4 Explanation STEP 1: Get rid of the fractions. by multiplying second equation by 5. After multiplying we have the following system: x+y=620; 5x =93 STEP 2: Use second equation to find x 5x=93x=93/5 STEP 3: Substitute x=93/5 into the first equation to solve for y. 93/5+y=620y=3007/5 Thus alcohol is 18.6 ml and water is 601.4 ml 30) Ans : Let x be quantity of one solution and y be of the second solution , now x+y=10; 3/20x+1/20y =11/10 The solutions are: x=6; y=4 Explanation STEP 1: Get rid of the fractions. by multiplying second equation by 20. After multiplying we have the following system: x+y =10; 3x+y =22 STEP 2: Multiply first equation by -3. After multiplying we have the following system: 3x3y =30; 3x+y =22 STEP 3: add the two equations together to eliminate x from the system. 2y=8 STEP 4: find y y=4 STEP 5: substitute the value for y into the original equation to solve for x. x+(4)=10 x=6 Thus 6 litre of 1st solution and 4 litre of wnd solution is needed 31) Ans: Let 1st investment is x and 2nd one is y Now ; x+y=5900; 11/100x+9/100y =593 The solutions are: x=3100; y=2800 Explanation STEP 1: Get rid of the fractions. by multiplying second equation by 100. After multiplying we have the following system: x+y=5900; 11x+9y =59300 STEP 2: Multiply first equation by -11. After multiplying we have the following system: 11x11y=64900; 11x+9y =59300 STEP 3: add the two equations together to eliminate x from the system. 2y=5600 STEP 4: find y y=2800 STEP 5: substitute the value for y into the original equation to solve for x. x+(2800)=5900 x=3100 Thus 1st investment is 3100$ and 2nd one is 2800 $ 32) Ans: OPTION B: Explaination : If the problem includes where it is equal, then you will have a solid boundary line. In other words, if you have < or > , you will have a solid line for your boundary line. Here it is < or > . Henxe solid line . 33) Ans: Yes Option A IS CORRECT . also one can check from given graph . The point (3,-4) lies in green area, so this is in the region 34) ANs : 2x-y<= 3 and x+y <=8 The colored region is solution 35) Ans : y+3x>=0 4x-y<= -1 The shaded region is solution 36) Ans : Shaded region is the answer 37) ANs : Shaded region is solution 38) ANs : Shaded region is solution 39) Ans: Shaded region is solution 40)Ans: : Shaded region is solution 41) Shaded region is solution These are parallel line , so inside region is solution .No solution on line . 1) Option C is correct. They don't intersect as they don't have solution ,Hence parallel lines 2) 2 solutions are possible here as 2 unknowns and 2 equation are there 3) Put (7/5 , 3 ) in both equations to check if this is solution a) LHS= 15(7/5)+20= 41 AND RHS= 44-3=41 So this satisifies b) LHS=5(7/5)-7(3)= -14 and LHS =4 not equal to LHS ,hence not a solution 4) Ans : A) The solution is (0,-4) it's intersecting point Graph method 10 8 6 4 Y Axis 2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 -2 -4 X Axis 4) 5) ans Option B graph is correct. Parralel lines as slope m1 and m2 are same m1=m2= -1 There is no solution 6)Ans Option B: there are infintely many solutions as equation 2 is same as 1, divide equation (2) by 3 , you get equation 1 . which are same Graph method 10 8 6 4 Y Axis 2 0 0 -2 -4 0.5 1 1.5 2 2.5 X Axis 3 3.5 4 4.5 7)Ans: substitute y=3x in equation (1) , we have 2x+3x=5 , 5x=5 thus x=1 , put this in equation 2 We get y=3 Thus (1, 3) is solution 8) Ans : Put y=4x-3 in equation (1), we get -x - 4(4x-3)= -5 -17x= -17 thus x=1 , put this in equation 1 , we get Y=1, Thus (1,1) is solution 9) Ans: Put equation (2) in equation (1), we have 3y +10 =(3/2)y ; Thus y= -20/3 , put this in eqn (2) X=-10 Thus ( -10, -20/3) is solution 10) Ans\" Put equation (1) in equation (2), we have 6x-3(2x)=0 thus 0=0 Equation 1 and 2 are same , divide equation (2) by 2 , we get y=2x same to equation 1. So there are infinite solution to this 11)Ans: Substitute x=6y in equation 2 , We get 24y-24y=4 , i.e. 0 = 4 , impossible Hence no solution possible 12) Ans Mulitiply equation 2 by 3 We have 9x+12y=48 Subtract equation 1 from equation 2 , We get , 5y=20 , thus y=4 , Put this in equation 1, we get , 9x+28=28 , Thus x=0 (0,4) is the solution Put this (0,4 ) in both equation, both satisfies . Hence checked, this is the solution 13) Ans : Multiply equation 1 by 2 ,we get 4s+10t =16 Now Add equation 1 from equation 2 ,we get 5s=39 , thus s= 5/39 Put this in equation 1 , we get 10/39 +5t=8 Thus t= 1.5487 Thus ( 5/39 , 1.5487) is solution Put this ( 5/39 , 1.5487) in both equation, both satisfies . Hence checked, this is the solution 14) Ans : Multiply equation 1 by 2/5 and equation 2 by 1/3 , we get 2/15 (x) - 1/30 (y) =2 2/15(x) +2/9 (Y) =2 .....(3) .....(4) Subtract equation 2 from equation 1, We get y=0 Put this y=0 in equation 3 to get x , X=15 , So (15, 0 ) is the solution Put this ( 15,0) in both equation, both satisfies . Hence checked, this is the solution 15) Ans: (2,3) is intersection point , hence this is solution Graph method 10 8.67 8 6 4 Y Axis 2 0 0 -2 -2 -0.67 0.5 0 1 1.5 2 2.5 3 3.5 4 4.5 -4 X Axis 16)Ans they are parallel lines( option B) , hence inconsistent equation s, hence no solution (option C) as they don't intersect Graph method 10 8 6 4 Y Axis 2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 -2 -4 X Axis 17 ANs : Divide equation 2 by -9 we get 8x-3y=6 , which is equal to equation 1 , It is the same line (Option B) , hence no solution (option C /)possible with two unknowns and 1 equation 18) Ans let small no. be x and big number be y Now x+y =103 ..... equation 1 And y- 3x =7 Subtract equation 2 from 1 , we get 4x=96 , thus x= 24 , Thus y= 103-24 =89 (24, 89 ) is solution Small number is 24 and larger one is 89 19) Ans Let heavy equi[ment operators be x and general laborers be y , Thus x+y= 37 ......equation (1) And 120x+86y=4304 ......equation (2) Multiply equation 1 by120 , we get 120x+120y=4440 Subtract equation 2 from this , we get 34y=136 , thus y=4 Now , x= 37-4 = 33 , Thus heavy equi[ment operators are 33 and general laborers are 4 20) ANs : Let regular seat be x and sleeper seat be y in number Thus x+y =96 ......equation (1) And 118x+295y=21063 ......equation (2) Multiply equation 1 by 118 , we get 118x+ 118y = 11328 , Subtract this from equation 2 we get 177y= 9735 thus y= 55 X=96-55 = 41 regular seat are 41 and sleeper seat be are 55 in no. 21) ANs : Let x be adults and y be children 3x+4y =79 multiply this by 2 , we have 6x+8y=158 2x+3y=56 6x+9y=168 multiply this by 3 , we have Subtract 1 from 2 , we have Y=10 , Thus put y=10 in equation 1 , we have 3x=79-40=39 Thus x= 13 We have 13$ for adults and 10 dollars for children 22) ANs: Let x be donuts and y be coffees 4x+5y=7.36 multiply by 6 , we have 24x+30y=44.16 6x+3y=6.9 multiply by 4 , we have 24x+12y=27.6 Subtract 2 from 1 , We have 18y= 16.56 Y=0.92 Now ,put in equation 1 , we have x=0.69 Cost of donut is 0.69$ and coffee is 0.92$ 23) Ans: let speed of plane be x and of wind be y The solutions are: x=3; y=1/2 Explanation STEP 1: Multiply each equation by a number that will create opposite coefficients for x. In this case we will multiply first equation by -5 and the second equation by 7. After multiplying we have the following system: 350x+350y=875; 350x+350y =1225 STEP 2: add the two equations together to eliminate x from the system. 700y=350 STEP 3: find y y=1/2 STEP 4: substitute the value for y into the original equation to solve for x. 70x70(12)=175 x=3 24) Ans : x+y =32 2x+y =51 The solutions are: x=19; y=13 Explanation STEP 1: Multiply first equation by -2. After multiplying we have the following system: 2x2y =64 2x+y =51 STEP 2: add the two equations together to eliminate x from the system. y=13 STEP 3: find y y=13 STEP 4: substitute the value for y into the original equation to solve for x. x+(13)=32 x=19 Thus 2 pointers are 19 and free throws are 13 25) Ans : Let calls on weekdays be x and on weekend be y x+y=593; 2/25x+9/100y =2575 The solutions are: X=197; y=396 Explanation STEP 1: Get rid of the fractions. by multiplying second equation by 100. After multiplying we have the following system: x+y=593; 8x+9y =5140 STEP 2: Multiply first equation by -8. After multiplying we have the following system: 8x8y=4744; 8x+9y =5140 STEP 3: add the two equations together to eliminate x from the system. y=396 STEP 4: find y y=396 STEP 5: substitute the value for y into the original equation to solve for x. x+(396)=593 x=197 Thus 197 calls on weekdays and 396 calls on weekends 26) Ans : Let games won be x and lost be y . x+y =70; xy=32 The solutions are: x=51; y=19 Explanation STEP 1: add the two equations together to eliminate y from the system. 2x=102 STEP 2: find x x=51 STEP 3: substitute the value for x into the original equation to solve for y. (51)+y=70 y=19 Games won are 51 and lost are 19 in total 27) Ans : Let x be length and y be breadth , 2x+2y =24; xy=2 The solutions are: x=7, y=5 Explanation STEP 1: Multiply second equation by -2. After multiplying we have the following system: 2x+2y=24; 2x+2y =4 STEP 2: add the two equations together to eliminate x from the system. 4y=20 STEP 3: find y y=5 STEP 4: substitute the value for y into the original equation to solve for x. 2x+2(5)=24 x=7 Thus length is 7 and breadth is 5 units 28) ANs : Let x be hot dog and y be hamburger , Now 7x+2y=232; 2x7y =714 The solutions are: x=11653; y=405212 Explanation STEP 1: Get rid of the fractions.by multiplying first equation by 2 and second equation by 4. After multiplying we have the following system: 14x+4y=23 8x28y =71 STEP 2: Multiply first equation by 7. After multiplying we have the following system: 98x+28y=161 8x28y =71 STEP 3: add the two equations together to eliminate y from the system. 106x=232 STEP 4: find x x=116/53=2.1887 STEP 5: substitute the value for x into the original equation to solve for y. 7(11653)+2y=232 y=405/212= 1.9104 thus hot dogs are 2 and hamburger are 2 29) ans : let x be alcohol and y be water in ml x+y=620; x=93/5 The solutions are: x=93/5 = 18.6 ; y=3007/5=601.4 Explanation STEP 1: Get rid of the fractions. by multiplying second equation by 5. After multiplying we have the following system: x+y=620; 5x =93 STEP 2: Use second equation to find x 5x=93x=93/5 STEP 3: Substitute x=93/5 into the first equation to solve for y. 93/5+y=620y=3007/5 Thus alcohol is 18.6 ml and water is 601.4 ml 30) Ans : Let x be quantity of one solution and y be of the second solution , now x+y=10; 3/20x+1/20y =11/10 The solutions are: x=6; y=4 Explanation STEP 1: Get rid of the fractions. by multiplying second equation by 20. After multiplying we have the following system: x+y =10; 3x+y =22 STEP 2: Multiply first equation by -3. After multiplying we have the following system: 3x3y =30; 3x+y =22 STEP 3: add the two equations together to eliminate x from the system. 2y=8 STEP 4: find y y=4 STEP 5: substitute the value for y into the original equation to solve for x. x+(4)=10 x=6 Thus 6 litre of 1st solution and 4 litre of wnd solution is needed 31) Ans: Let 1st investment is x and 2nd one is y Now ; x+y=5900; 11/100x+9/100y =593 The solutions are: x=3100; y=2800 Explanation STEP 1: Get rid of the fractions. by multiplying second equation by 100. After multiplying we have the following system: x+y=5900; 11x+9y =59300 STEP 2: Multiply first equation by -11. After multiplying we have the following system: 11x11y=64900; 11x+9y =59300 STEP 3: add the two equations together to eliminate x from the system. 2y=5600 STEP 4: find y y=2800 STEP 5: substitute the value for y into the original equation to solve for x. x+(2800)=5900 x=3100 Thus 1st investment is 3100$ and 2nd one is 2800 $ 32) Ans: OPTION B: Explaination : If the problem includes where it is equal, then you will have a solid boundary line. In other words, if you have < or > , you will have a solid line for your boundary line. Here it is < or > . Henxe solid line . 33) Ans: Yes Option A IS CORRECT . also one can check from given graph . The point (3,-4) lies in green area, so this is in the region 34) ANs : 2x-y<= 3 and x+y <=8 The colored region is solution 35) Ans : y+3x>=0 4x-y<= -1 The shaded region is solution 36) Ans : Shaded region is the answer 37) ANs : Shaded region is solution 38) ANs : Shaded region is solution 39) Ans: Shaded region is solution 40)Ans: : Shaded region is solution 41) Shaded region is solution These are parallel line , so inside region is solution .No solution on line